I then saved it to the default plugin directory: C:\Users\~\AppData\Roaming\Sublime Text\Packages

2) I then went to Preference > User Key Bindings, and pasted in the command I found here like this:

<!--
Place your key bindings in here, this will ensure they don't get overwritten
when installing new versions of Sublime Text
-->
<bindings>
<binding key="ctrl+period" command="closeTag"/>
</bindings>

Hitting Ctrl + . does not do anything though. And nothing shows up in the console when I do.

For the purpose of this plugin, it will work the majority of the time. It's meant to be called manually so I think it's more than good enough.

But if you know of a good parser than can be easily integrated, it could lead to for interesting plugins. The parsers I've used are meant to extract information: they return a tree that doesn't have the character position of the nodes in the document.